The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules i2=a{\displaystyle i^{2}=a}, j2=b{\displaystyle j^{2}=b}, ij=−ji=k{\displaystyle ij=-ji=k}. In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.

For a placev of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completionQv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field.

Over the reals, (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative.

Over the p-adics with p odd, writing a=pαu{\displaystyle a=p^{\alpha }u} and b=pβv{\displaystyle b=p^{\beta }v}, where u and v are integers coprime to p, we have

where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F.[1]

The radical is a subgroup of F*/F*2, identified with a subgroup of F*. The radical is equal to F* if and only if F has u-invariant at most 2.[2] In the opposite direction, a field with radical F*2 is termed a Hilbert field.[3]

If K is a local field containing the group of nth roots of unity for some positive integer n prime to the characteristic of K, then the Hilbert symbol (,) is a function from K*×K* to μn. In terms of the Artin symbol it can be defined by[4]

Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for n prime) used the power residue symbol when K has residue characteristic coprime to n, and was rather complicated when K has residue characteristic dividing n.

Hilbert's reciprocity law states that if a and b are in an algebraic number field containing the nth roots of unity then[5]

∏p(a,b)p=1{\displaystyle \prod _{p}(a,b)_{p}=1}

where the product is over the finite and infinite primes p of the number field, and where (,)p is the Hilbert symbol of the completion at p. Hilbert's reciprocity law follows from the Artin reciprocity law and the definition of the Hilbert symbol in terms of the Artin symbol.

If K is a number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then the power residue symbol (ap) is related to the Hilbert symbol by[6]

(ap)=(π,a)p{\displaystyle {\binom {a}{p}}=(\pi ,a)_{p}}

The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field
by putting (ab)=(a(b)) where (b) is the principal ideal generated by b.
Hilbert's reciprocity law then implies the following reciprocity law for the residue symbol, for a and b prime to each other and to n: